Spreadsheets are a valuable
resource for analysis of most kinds of data in sport and exercise
science.Here I present a
spreadsheet for comparison of change scores resulting from a treatment in
an experimental and control group. Features of the spreadsheet include: the
usual analysis based on the unequal-variances unpaired t statistic;
analysis following logarithmic, percentile-rank, square-root, and
arcsine-root transformations; plots of change scores to check for
uniformity of the effects; back-transformation of the effects into
meaningful magnitudes; estimates of reliability for the control group;
estimates of individual responses; comparison of the groups in the pre-test;
and estimates of uncertainty in all effects, expressed as confidence limits
and chances the true value of the effect is important.Analysis of straightforward crossover
trials based on the paired t statistic is provided in a modified version of
the spreadsheet. KEYWORDS: analysis, crossover, design, intervention,
psychometric, randomized, statistics, transformation, t test.

Amongst the various kinds of research
design, a controlled trial is the best for investigating the effect of a
treatment on performance, injury or health (Hopkins, 2000a). In a controlled
trial, subjects in experimental and control treatment groups are measured at
least once pre and at least once during or post their treatment.The essence of the analysis is a comparison
between groups of an appropriate measure of change in a dependent variable
representing performance, injury, or health. The final outcome statistic is
thus a difference in a change: the difference between groups in their mean
change due to the experimental and control treatments.Another form of controlled trial is the
crossover, in which all subjects receive all control and experimental
treatments, with sufficient time following each treatment to allow its effect
to wash out. The final outcome statistic in a crossover is simply the mean
change between treatments.

Calculating the value of the outcome
statistic in controlled trials and crossovers is easy enough.More challenging is making inferences about
its true or population value in terms of a p value, statistical significance,
confidence limits, and clinical significance.The traditional approach is repeated-measures analysis of variance
(ANOVA).In a slideshow I presented
at a conference recently, I pointed out how this approach can lead to the
wrong p value and therefore wrong conclusions about the effect of the
treatment in controlled trials. I also
explained how the more recent approach of mixed modeling not only gives the
right answers but also permits complex analyses involving additional levels
of repeated measurement in addition to covariates for subject
characteristics.Mixed modeling is
available only in advanced, expensive, and user-unfriendly statistical
packages, such as the Statistical Analysis System (SAS), but straightforward
analysis of controlled trials either by repeated-measures ANOVA or mixed
modeling is equivalent to a t test, which can be performed on an Excel
spreadsheet.

In the last few years I have been advising
research students to use a spreadsheet for their analyses whenever possible
and to consult a statistician only when they need help with more complex
analyses. Spreadsheets I have devised for various analyses (reliability, validity, assessment of an individual,
confidence limits
and clinical significance) seem to have facilitated this process.Although it may be instructive for students
to devise a spreadsheet from scratch, they probably reach a more advanced
level more rapidly by using a spreadsheet that already works: anytime they
want to learn more about the calculations, they need only click on the
appropriate cells.Use of a
pre-configured spreadsheet surely also reduces errors and saves more time
(the student's and the statistician's).I have therefore devised a spreadsheet for analysis of straightforward
controlled trials, and I have modified it for analysis of crossovers.

Controlled
Trials

Features of the spreadsheet for controlled
trials include the following…

•The usual analysis of the raw
values of the
dependent variable, based on the unequal-variances unpaired t statistic.

The
data in the spreadsheet are for one pre- and two post-treatment trials, and
the effects are the differences in the three pairwise changes between the
trials (post1-pre, post2-pre, and post2-post1).You can easily add more trials and effects,
including parameters for line or curve fits to each subject's trials–what I
call within-subject
modeling (Hopkins, 2003a).As for
the role of the unequal-variances t statistic, see the section on uniformity of residuals
at my stats site (Hopkins, 2003b) and also the slideshow I referred to
earlier. In short, never use the equal-variances t test, because the variances
are never equal. (The variances in question are the squares of the standard
deviations of the change scores for each group.)

With
three or more groups (for example, a control and two treatment groups), you
will have to use a whole new spreadsheet for each pairwise comparison of
interest.Enter the data for two
groups, save the spreadsheet, resave it with a new name, then replace one
group's data with those of the third group.Save, resave, then copy and paste data and so on until you have all the
pairwise group comparisons.

The
spreadsheet does not provide any adjustment for so-called inflation of Type 1 error
with multiple group comparisons or with the multiple comparisons between more
than two trials (Hopkins, 2000b).These adjustments (Tukey, Sidak, and so on) probably don't work for
repeated measures, and in any case they are nonsense, for various reasons
that I detail at my stats site and in that slideshow.The main reason is that the procedure,
which involves doing pairwise tests with a conservatively adjusted level of
significance only if the interaction term is significant, dilutes the power
of the study for the most important comparison (for example, the last pre
with the first post for the most important experimental vs control
group).I don't believe in testing for
significance anyway, but even if I did, I would be entitled to apply a t test
to the most important pre-planned comparison without looking at the
interaction and without adjustment of significance for multiple comparisons.
Some of the best researchers in our field fail to understand this important
point.

•Analysis of various transformed
values of the
dependent variable, to deal with any systematic effect of an individual's
pre-test value on the change due to the treatment.

For
example, if the effect of the treatment is to enhance performance by a few
percent regardless of a subject's pre-test value, analysis of the raw data
will give the wrong answer for most subjects.For these and most other performance and physiological variables,
analysis of the logarithm of the raw values gives the right answer.Along with logarithmic transformation, the
spreadsheet has square-root
transformation for counts of injuries or events, arcsine-root transformation
for proportions, and percentile-rank
transformation (equivalent to non-parametric analysis) when an
appropriate formula for a transformation function is unclear or unspecifiable
(Hopkins, 2003c and other pages).

A
dependent variable with a grossly non-normal distribution of values and some
zeros thrown in for good measure is a good candidate for rank
transformation.An example is time
spent in vigorous physical activity by city dwellers: the variable would
respond well to log transformation, were it not for the zeros.

Dependent
variables with only two values (example: injured yes/no) and Likert-scale variables
with any number of points (example: disagree/uncertain/agree) can be coded as
integers and analyzed directly without transformation (Hopkins, 2003b).I now code two-value variables and 2-point
scales as 0 or 100, because differences or changes in the mean then directly
represent differences or changes in the percent of subjects who, for example,
got injured or who gave one of the two responses.Advanced approaches to such data involve
repeated-measures logistic regression, but the outcomes are odds ratios or
relative risks, which are hard to interpret.

If
you have a variable with a lower and upper bound and values that come close
to either bound, consider converting the values so that they range from 0 to
100 ("percent of full-scale deflection"), then applying the arcsine-root
transformation. Composite psychometric scores derived from multiple Likert
scales should behave well under this transformation, especially when a
substantial proportion of subjects respond close to the minimum or maximum
values.

•Plots of change scores of raw and transformed data
against pre-test values, to check for outliers and to confirm that the chosen
transformation results in a similar magnitude of change across the range of
pre-test values.

These
plots achieve the same purpose as plots of residual vs predicted
values in more powerful statistical packages. Statisticians justify
examination of such plots by referring to the need to avoid
heteroscedasticity (non-uniformity of error) in the analysis, which for a
controlled trial means the same thing as aiming for uniformity in the effect
of the treatment.

Sometimes
it's hard to tell which transformation gives the most uniform effect in the
plots.Indeed, when there is little
variation in the pre-test values between subjects, all transformations give
uniform effects and the same value for the mean effect after back
transformation.Regardless, your
choice of transformation should be guided by your understanding of how a wide
variation in pre-test values would be likely to affect the effects.

After
applying the appropriate transformation, you may sometimes still see a
tendency for subjects with low pre-test values to have more positive change
scores (or less negative change scores) than subjects with high pre-test
values. This tendency may be a genuine effect of the pre-test value, but it
may also be partly or wholly an artefact of regression to the mean (Hopkins,
2003d).You address this issue by
performing an analysis of variance or general linear model with the change
score as the dependent variable, group identity as a nominal predictor, and
the pre-test value as a numeric predictor or covariate interacted with group.The difference in the slope of the
covariate between the two groups is the real effect of pre-test value free of
the artefact.

•Various solutions to the problem
of back-transformation of treatment effects into meaningful magnitudes.

Log-transformation
gives rise to percent effects after back transformation.If the percent effects or errors are large
(~100% or more, as occurs with some hormones and assays for gene expression),
it is better to back-transform log effects into factors.For example, an increase of 250% is better
expressed as a factor of 3.5.

Another
approach, which as far as I know is novel, is to estimate the value of the
effect at a chosen value of the raw variable. I have included this approach
for back-transformation from percentile-rank, square-root, and arcsine-root
transformations.See the spreadsheet
to better understand what I mean here.Note that I have not included this approach with log transformation,
because percent and factor effects are better ways to back transform log effects.

Finally,
I have also expressed magnitudes of effects for the raw variable and for all
transformations as Cohen effect
sizes:the difference in the
changes in the mean as a fraction or multiple of the pre-test between-subject
standard deviation.You should
interpret the magnitude of the Cohen effect sizes using my scale: <0.2 is
trivial, 0.2-0.5 is small, 0.6-1.1 is moderate, 1.2-1.9 is large, and 2.0 or
more is very large (Hopkins, 2002a).Use Cohen effect sizes for effects that relate to population health,
for effects derived from physiological, biomechanical, or other mechanism
variables, and for some measures of performance that have no direct
association with competitive performance scores (for example, field tests for
team-sport athletes). Do NOT use Cohen effect sizes for direct measures of
competitive athletic performance or performance tests directly related
thereto; instead express the effects as percents or as raw units.

•Estimates of reliability in the control group, expressed
as typical error of measurement and change in the mean.

The
design and analysis of the control group on its own amount to a reliability study. You can
therefore compare measures of reliability derived from the control group with
those of published reliability studies or your own pilot reliability study,
if you did one. The most important measure of reliability is the typical
error: the typical variation (standard deviation) a subject would show in
many repeated trials (Hopkins, 2000c and my stats pages on reliability). The typical
error shown in the spreadsheet is simply the standard deviation of the change
score in the control group divided by Ö2.If your error differs substantially from that of previous studies, you
should try and explain why in your write-up.Note that one obvious explanation for a worse error in your study is a
longer time between trials compared with that in reliability studies.

The
change in the mean from trial to trial is another measure of reliability. A
substantial change usually indicates that the subjects (or the researcher)
showed a practice, learning, or other familiarization effect between the
trials.Again, you will need to
explain any such change in your study. A substantial change in the mean can
also help explain a worse typical error than in previous studies:there are usually individual responses to
the change (some subjects show more familiarization), and individual
responses produce more variation in change scores.

•Estimates of individual responses to the treatment, expressed as a
standard deviation for the mean effect, in the various forms of the
transformed and back-transformed variable.

If
subjects differ in their response to the experimental treatment after any
appropriate transformation, the standard deviation of the change scores in
the experimental group will be greater than that in the control group.It is easy to show from basic statistical
theory that the variation in the response between individuals, expressed as a
standard deviation, is the square root of the difference in the variances
(square of the standard deviations) of the change scores.This estimate of variation is free of error
of measurement, although error of measurement obviously contributes to its
uncertainty.

The
standard deviation representing individual responses is the typical variation
in the response to the treatment from individual to individual. For example, if the mean response is 3.0
units and the standard deviation representing individual responses is 2.0
units, most individuals (about two-thirds) will have a true response
somewhere in the region of 1 to 5 (3-2 to 3+2). You can state that the effect
of the treatment is typically 3.0 ± 2.0 units (mean ± standard
deviation).

Sometimes
the observed difference in the variances of the change scores is negative,
either because the experimental treatment somehow reduces variation between
subjects (for example, by bringing them up to a common ceiling or plateau),
or more likely because sampling variation results in a negative difference
purely by chance.I have devised the
convention of showing the square root of a negative variance as a negative
standard deviation.You should
interpret such negative standard deviations as indicative of no individual
responses.

If
you find that there are substantial individual responses to a treatment, the
next step is to find the subject characteristics that predict them. To do
that properly, you have to include subject characteristics in the analysis as
covariates, which is not possible with the spreadsheet.However, you can do a publication-worthy
job by correlating or plotting the change scores with the subject characteristics
that you think might predict the individual responses.See the slideshow on repeated
measures for more information on individual responses, including a caution
about misinterpreting a poor correlation between change scores.

•A comparison of pre-test values of means and standard deviations
in the experimental and control groups for all transformations and
back-transformations, to check on the balance of assignment of subjects to the
groups.

The
purpose of a control group is to provide an estimate of the change that
occurs in the absence of the experimental treatment. This change is then
subtracted from the change in the experimental group to give the pure effect
of the experimental treatment. Fine, but suppose the pre-test score affects
the change resulting from the experimental and control treatments (example:
subjects with lower pre-test scores respond more to either or both
treatments).Suppose also that the
mean pre-test score differs substantially between the groups–that is, the
groups are not balanced with respect to the pre-test score. Under these
circumstances, part of the difference between the change scores will be due
to the lack of balance, so the analysis will not provide a correct estimate
of the experimental effect.It is
therefore important to compare the two groups and comment on the difference
in the mean pre-test score.The
spreadsheet provides you with such a comparison.You can describe the magnitude of the difference
between the groups either by noting whether the difference is greater than
the smallest worthwhile change (see the last bullet point below) or by
interpreting the Cohen effect size for the difference as trivial, small, and
so on, when Cohen is appropriate. It is equally important to compare the
group mean values of any other subject characteristic that could interact
with the two treatments (examples: age, competitive level, proportion of
males…), but you will have to re-jig the spreadsheet for such comparisons
yourself.

When
there is a substantial difference between the groups in the pre-test
mean value of the dependent variable or of another subject characteristic,
make sure you examine the plots of the change scores against pre-test value or
create plots to examine the effect of the other subject characteristic on the
change scores.If you see a
substantial effect of pre-test value on the change scores, perform the
analysis of variance or general linear model I described in the section on plots
of change scores to deal with regression to the mean.That way you will have adjusted
statistically for the differences in the groups in the pre-test.Your stats package should give you the
experimental effect as the difference in the change for subjects who would be
on the overall mean value of the covariate, because this value corresponds to
the expected value for the population represented by the full sample. You
might need an expert to help you here.

You
can also use this part of the spreadsheet for a comparison of means (and
standard deviations) of any two independent groups without repeated
measurements.Just ignore all the
analyses related to changes in the means.

•Estimates of uncertainty expressed as confidence limits at
any percent level for all effects, including the standard deviations
representing individual responses.

Confidence limits
define a range within which the true value (that is, the population value) of
something is likely to occur, where likely is traditionally with
95% certainty.I now favor 90%
confidence limits, because I believe 95% limits give an impression of too
much uncertainty.They are also too
easily reinterpreted in terms of statistical significance at the 5% level,
which I am also keen to avoid.

The
confidence limits for the treatment effects are generated using the same
formulae as in my spreadsheet
for confidence limits.Confidence
limits for the standard deviation representing individual responses are based
on one of the methods used in Proc Mixed in SAS.The sampling distribution of the difference
of the variances of the change scores is assumed to be a normal
distribution.The variance of the
sampling distribution of a variance is 2(variance)2/(degrees of
freedom).The variance of the
difference in the variances is simply the sum of the two sampling variances,
because the control and experimental groups are independent.

I
have supplied confidence limits for the comparison of the two groups in the
pre-test, but I don't think we should use them for controlled trials.After all, any difference between the two
groups is real as far as imbalance in the assignment is concerned.What matters is how different the groups were
for the study, not how different the groups could be if we drew a huge
sample to get the difference between the population values for the control
and experimental groups. But if you are using the spreadsheet only for a
comparison of means and standard deviations of a single measurement of
subjects in two groups, then of course confidence limits are appropriate.

•Chances that the true value of an
effect is important,
for all comparisons of means.

Important means worthwhile or substantial
in some clinical, practical, or mechanistic sense.You provide a value for the effect that you
consider is the smallest that would be important for your subjects.The spreadsheet then estimates the chances
that the true value is greater than this smallest important value, and it
also shows the chances in a qualitative form (unlikely, possible,
almost certain, and so on).

Sometimes
you have no clear idea of the smallest important value, especially for
physiological or other mechanism variables.In such cases, use Cohen's smallest effect size of 0.2, which is
included in the spreadsheet as the default.By inserting values of 0.6, 1.2, or 2.0, you can also estimate the
chances that the true value is moderate, large, or very large.Try these different values when you want to
say something positive about a smallish effect that has considerable
uncertainty arising from large error of measurement, individual responses, or
a small sample size.You can usually
state something like the mean effect could be trivial or small, but it is
unlikely to be moderate and is almost certainly not large.Such statements will help get your
otherwise apparently inconclusive study into a journal, if the reviewers and
editor are enlightened.

The example in the spreadsheet is for a
control treatment and two experimental treatments, with equal numbers of
subjects on each of the six possible treatment sequences.For a study with a single experimental
treatment, ignore or delete the columns relating to the second experimental
treatment.Add more columns to add
more treatments. You can also use the spreadsheet to analyze a time series,
in which all subjects receive the same treatment sequence. This design
obviously isn't as good as a crossover, but it can still provide valuable
information if you do multiple trials before, during and/or after the
treatment.

The analyses are based on the paired t
statistic.For convenience, the t
statistic is generated by comparing a column of change scores with a column
of zeros rather than by comparing directly the columns of scores for each
treatment. This approach also allows you to include and analyze a column
representing any effect derived from within-subject modeling, such as a slope
representing a gradual change in a time series, or parameters describing a
dose-response relationship when the treatments differ only in dose.

Plots of change scores in a crossover fulfill
the same roles as in a controlled trial: to check for outliers and to check that
your chosen transformation makes the effect reasonably uniform across the
range of subjects' values for the control treatment (the equivalent of the
subjects' values for the pre-test in a controlled trial). Beware that
regression to the mean will sometimes be responsible partly or entirely for
any trend towards experimental-control changes that get more positive for
lower control scores, even in the plots with the most appropriate
transformation. To see the effect of the control value on the experimental
treatment free of this artifact, one of the treatments in the crossover needs
to be a repeat of the control treatment.The plot of the experimental-control change scores for one control
against the values for the other control shows the pure effect of the control
value on the experimental treatment.Fit a line or a curve to the points to quantify the effect, or do an
equivalent analysis with mixed modeling.

Lack of a control group in a crossover
precludes estimation of reliability, but I have included estimates of the
typical error derived from the changes between treatments.This error is still formally a measure of
reliability.If you compare your
values with those from reliability studies, take into account the possibility
that individual responses to one or more of your treatments can inflate the
error.

Another potential source of inflation of
error in a crossover is a substantial familiarization effect, particularly
between the first and second trials. This effect adds to the change scores of
subjects who have the control treatment first, whereas it subtracts from
those who have it second.If there are
equal numbers of subjects in each treatment sequence, there is no nett effect
on the mean change score (the treatment effect), but the random error
increases.Analysis with mixed
modeling allows you to estimate the magnitude of the familiarization effect
and to eliminate this source of error by including the order of the
treatments as a within-subject nominal covariate in the fixed-effects
model.You end up with a more precise
estimate (narrower confidence interval) for the treatment effect.Mixed modeling also provides a check on the
balancing of the assignment.Get your
subjects to repeat the control treatment, preferably in a balanced manner, and
it will also provide you with estimates of reliability and individual
responses.

In conclusion, the spreadsheets should help
you analyses your data appropriately.Admittedly, you can't include covariates in the analysis, but the
spreadsheet should still help you do the right things when you use a more
sophisticated statistical package.